Fujji [1]} and Simon [2]} verified
that \(\sigma ^{\ast }\)  is bounded from \(H_{1}\)  to \(L_{1}\) . Weisz [3]}
generalized this result and proved boundedness of \(\sigma ^{\ast }\)  from the
martingale space \(H_{p}\)  to the Lebesgue space \(L_{p}\)  for \(p>1/2\) . Simon
[4]} gave a counterexample, which shows that boundedness does not hold
for \(0<p<1/2.\)  A counterexample for \(p=1/2\)  was given by Goginava [5]} (see also [6]} and [7]}). Weisz [8]} proved that
the maximal operator of the FejĂŠr means \(\sigma ^{\ast }\)  is bounded from
the Hardy space \(H_{1/2}\)  to the space \(weak-L_{1/2}\) . The boundedness of
weighted maximal operators are considered in [9]},
[10]} and [11]}.
